lecture07

lecture07 - Yinyu Ye, MS&E, Stanford...

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Unformatted text preview: Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #07 1 SensitivityAnalyses YinyuYe DepartmentofManagementScienceandEngineering StanfordUniversity Stanford,CA94305,U.S.A. http://www.stanford.edu/yyye Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #07 2 ParametricLinearProgrammingProblem The objectivecoefficient vectorbecomes c + g and/orthe right-hand-side vectorbecomes b + d ,wheretheparameter belongstoan interval . DenotethisproblembyLP( ): LP ( ) minimize ( c + g ) T x subjectto A x = b + d , x . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #07 3 GeometricalObservations 1.Weknowthatforthefunction c T x ,thevector c denotesthe directionof steepestascent .Thus, parameterizing thecostfunctionaccordingtotherule c + g changesthe gradientorslope oftheobjectivehyperplane.This raisesthequestionofwhetherornotthecurrent basis B remainsoptimal . 2.If b isreplacedby b + d ,as varyingthepoint b + d movesawayfrom b inthedirection d (dependingonthesignof ).Thisraisesthequestionof whetherornotthenewRHS b + d hasthecurrent basis B remains feasible ,thatis, A B x B = b + d implies x B . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #07 4 RecallWhenaBSFisoptimal GivenaBFSintheLPstandardform A B x B = b , x B , x N = , anditscompanionshadowprice(dualsolution)andreducedcostvectors: A T B y = c B , ( and r = c- A T y ) . Ifthereducedcostvector r ,thentheBFSwithbasicvariableset B ,thenthe BFSis optimal .(Whataboutthe maximization ?) Atoptimalitywealwayshave c T x = b T y . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #07 5 GettingStarted Letusconsider varyingaround . Akeyquestionintheseparametricproblemsis:howmuchcantheparameter bechangedbeforethecurrent optimalbasis B ofLP( )islost? Theorem1 TheoptmalbasisofLP( )remainsoptimalforLP( )ifandonlyif A- 1 B ( b + d ) and ( c + g )- A T ( A TB )- 1 ( c + g ) B . Thiswillestablishanintervalon...
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lecture07 - Yinyu Ye, MS&E, Stanford...

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